3.  Berry-Esseen-type bounds for Gaussian approximations and FDR control.
Our goal is to estimate the cluster-average and latent graphs, respectively, and to provide guarantees on the recovery error. Existing literature focuses on constructing
approximate confidence intervals for one or a finite number of entries of a CI graph, or known linear functionals of such entries [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}. In all these cases,
deriving the asymptotic limiting distribution of appropriate test statistics suffices. Our focus here is on the estimation of the sparsity pattern, which can be equivalently viewed
as a multiple-testing problem. It is well known that the exact sparsity pattern can be recovered, with high probability, only if the entries of each precision matrix are above the minimax optimal noise level \(O(\sqrt{\log d/n})\)  [12]}, [13]}. Since our aim is inference on the sparsity pattern without further restrictions on the parameter space, the next best type of error that we can control is the False Discovery Rate (FDR) [14]}. For this, we first derive Gaussian approximations for the distribution of our estimates. We establish Berry-Esseen type bounds on the difference between the cumulative distribution function of our estimators and that of a standard Gaussian random variable that are valid for each \(K, d\)  and \(n\) , and are presented in Theorems REF  and REF , respectively. In Section REF  we use these results for pattern recovery under FDR control, and explain the effect of the asymptotic approximations on this quantity.
